FDTD Analysis of Distribution Line Voltages Induced by Inclined … · 2017-05-09 · FDTD Analysis...
Transcript of FDTD Analysis of Distribution Line Voltages Induced by Inclined … · 2017-05-09 · FDTD Analysis...
FDTD Analysis of Distribution Line Voltages
Induced by Inclined Lightning Channel
Masashi Natsui, Akihiro Ametani, Jean Mahseredjian, Shozo Sekioka, Kazuo Yamamoto
Abstract--This paper investigates lightning induced voltages
on a distribution line, when the lightning channel is not vertical
but angled to the line and the earth representation is based on
the finite-difference time-domain (FDTD) method. The effect of
the current flowing into the earth, after the lightning channel
touches the earth, is also investigated. The induced voltages are
quite dependent on the angle of the channel to the line as can be
easily estimated. It is found that the lightning angle with the line
causes significant increase of the voltage and the effect of the
earth resistivity on the induced voltage has been made clear
when considering the inclined channel. Since the induced voltage
is, in theory, proportional to the frequency of the inducing
current, i.e. inversely proportional to rise time Tf when the
inducing and induced circuits are parallel. When the lightning
channel is angled, the proportional relation is less clear than that
in the vertical channel. This fact has to be taken into account
when discussing the effect of Tf. The current flowing in the earth
is also affected by the lightning inclination and the voltage drop
along the earth surface due to the current becomes larger than
that in the vertical lightning case.
Keywords: lightning induced voltage, distribution line, angled
channel, earth current, FDTD.
I. INTRODUCTION
OWER quality issues are becoming more and more
significant, especially in the field of distribution systems,
because a number of electronic devices, which are very
sensitive to overvoltages, are installed in the distribution
systems including home appliances [1]-[8]. Lightning to
ground nearby the distribution systems produces induced
lightning surges as is well-known [9]-[14]. An accurate
evaluation of lightning induced voltages is, therefore, essential
to protect these devices. In the induced surge calculations on a
distribution line, the lightning channels is, in most cases,
assumed to be straight and vertical to the ground surface and
the distribution line, because existing analysis techniques are
based on a transmission line (TL) theory which cannot handle
non-uniform conductors in a three-dimensional space [15]-
[19]. Also, a perfectly conducting ground is quite often
assumed in a calculation due to the theoretical premise. In
reality, a lightning channel is not straight nor vertical, and its
return stroke current flows into a ground which is not perfectly
conducting. The assumptions could result in incorrect
M. Natsui, A. Ametani, and J. Mahseredjian are with Polytechnique
Montreal, 2500, chemin de Polytechnique, Montréal (Québec) H3T1J4,
Canada (e-mail: [email protected])
S. Sekioka is with Shonan Institute of Technology, Fujisawa, Kanagawa,
Japan, 251-8511
K. Yamamoto is with Chubu University, Kasugai, Aichi, Japan, 487-8501
Paper submitted to the International Conference on Power Systems
Transients (IPST2017) in Seoul, Republic of Korea June 26-29, 2017
lightning induced surges. Therefore, these assumptions require
a further careful investigation for the design of effective and
adequate lightning protection methodologies.
In this paper, the effect of inclined lightning channel on a
lightning induced voltage on a distribution line is investigated
considering the earth resistivity by using a 3D finite-difference
time-domain (FDTD) method [20]-[22]. The distribution line
with finite length is placed above a lossy ground with an
orthogonal three-dimensional space. The induced voltage is
calculated at the position of the line closest to the base of the
lightning channel.
The FDTD modeling of each element involved in a model
circuit are described in Chapter II. FDTD computed results
under various conditions of a lightning channel together with
the return stroke velocity of the channel are shown, and the
effect of the inclined lightning channel on the induced voltage
is investigated in Chapter III. The contribution of the earth
surface current to the induced voltage is discussed in Chapter
IV. The remarks found in the investigations are summarized in
Chapter V.
II. MODEL CIRCUIT
In FDTD lightning study, lightning is generally modeled as
a combination of a thin wire and a current source on the bottom
of the lightning channel. In this paper, the lightning channel is
represented by traveling-current-source model [22] to adjust
the velocity and to obtain a smooth electromagnetic field.
Fig. 1 illustrates the model. The current source array is
aligned vertically and horizontally along the arbitrary inclined
lightning path and the velocity is adjusted by the delay time of
the excitation for each current source. The height of the
lightning channel is set to 500 m, and two directions of
inclined channel are investigated: angle θ on yz plane for the
inclination toward the line, and angle φ on xz plane for the
inclination along the line. The angle θ and φ vary from ̶ 45˚ to
+90˚ and 0˚ to +90˚, respectively. The angle of +90˚ is used for
the confirmation of consistency. Also, two cases of return
stroke velocity, β = 1.00 and 0.33, and two values of the wave
front duration, Tf = 1.0 and 0.1 μs are adopted in this study.
Note that β is the ratio of the return stroke velocity to the light
speed. A single overhead line with a height of 10 m is placed
at 50 m away from the lightning base, which represents a 20
kV distribution line. In general, when the fast-rise return
stroke current and low earth resistivity are assumed, the
induced voltage becomes proportional to the line height and
almost inversely proportional to the distance from the
lightning base [9], [23].
Both ends of the line are connected to Liao’s absorbing
boundary of the working space of 1060 m × 1060 m × 600 m,
P
which is composed of cubic cells of the several sizes between
1 m × 1 m × 1 m and 8 m × 8 m × 8 m. The minimum grid
size is adopted in the area including the lightning channel base
and the measuring point (140 m ×140 m × 140 m). The
measuring point of the induced voltage on the line is placed at
the closest point from the lightning base. The depth of earth is
taken as 100 m, and the earth resistivity ρe is set to 0 Ωm
(perfectly conducting earth: PCE), 100 Ωm, and 2000 Ωm. A
normalized value of current 1A with Tf = 1.0 and 0.1 μs shown
in Fig. 2, is adopted for each current source as a lightning
return stroke current.
III. LIGHTNING INCLINATION
A. Effect of angle θ
Fig. 3 shows induced voltage waveforms on a distribution
line for various values of θ. In this case (a) is for the perfectly
conducting earth, (b) is for earth resistivity ρe = 100 Ωm, and
(c) for ρe = 2000 Ωm. It is clear that for θ = ̶ 45˚, i.e. the
channel lightning channel being the nearest to the line, induces
the highest voltage to the line. As θ increases, i.e. the channel
becomes further to the line, the induced voltage tends to
decrease except the case of (c) ρe = 2000 Ωm as shown in Fig.
4. When ρe = 2000 Ωm, the induced voltages in the cases of θ
= ̶ 30˚ and 0˚ are higher than that in the case of θ = ̶ 45˚.
It is observed in Fig. 4 that the induced voltage becomes
higher as the earth resistivity increases as is well known [24]-
[25]. Also, it should be noted that the rise time of the induced
voltage becomes larger as ρe increases, and thus the induced
voltage sustains much longer in the case of ρe = 2000 Ωm than
that in the case of ρe = 100 Ωm. When θ = +90˚, the lightning
channel is placed above the earth surface by the height of 1 m,
i.e. the channel is almost perpendicular to the distribution line,
and thus the induced voltage is the smallest. For ρe = 0 Ωm,
the voltage is almost zero, and increases to 5.5V for ρe = 2000
Ωm. Considering the fact that the channel is perpendicular to
the line for θ = +90˚ is considered, the induced voltage may be
caused by the current flowing on the earth surface after the
lightning hits the earth. The characteristic of the current will
be discussed in Chapter IV as the earth surface current.
B. Effect of angle φ
Fig. 5 shows induced voltage waveforms with various
angles φ for ρe = 0, 100 and 2000 Ωm. Fig. 6 shows the
maximum voltage as a function of φ. The induced voltages
become higher as ρe increases. The highest voltage is observed
when φ = +45˚ and the voltage decreases as φ becomes smaller
except φ = +90˚. When φ = +90˚, the induced voltage becomes
almost zero for ρe = 0 Ωm, nearly the same as that of φ = +45˚
for ρe = 100 Ωm, and much higher for ρe = 2000 Ωm. The
phenomena are estimated to be caused by the image current in
the earth. When φ = +90˚, the lightning channel is also placed
1 m above the earth surface. Therefore, the image current
becomes large under the channel for ρe = 0 Ωm, and it cancels
almost all the electromagnetic field generated from the
lightning channel. Therefore, nearly no voltage is induced on
the line. On the other hand, the effect of the image current
becomes less for large ρe, and the electromagnetic field can
reach the line and induces the voltage.
C. Effect of rise time Tf
Fig. 7 shows a comparison of the results for Tf = 0.1 μs and
1.0 μs at angle θ = ̶ 45˚. The first peaks of the induced voltage
become larger and sharper due to the faster rise time of the
lightning current.
The relation between the peak voltage and angle θ for Tf =
0.1 μs, shown in Fig. 8 (a), is also similar to the result for Tf =
1.0 μs. The peak voltage increases as angle θ increases. The
increase ratio of the voltage according to angle θ is larger than
those for Tf = 1.0 μs, indicating that the induced voltage is
more sensitive to the angle in shorter Tf. Note that the induced
voltage is generally proportional to the frequency of the
inducing current, i.e. inversely proportional to Tf when the
inducing and induced circuits are parallel. However, the
induced voltage, which is caused by the superposition of
the electromagnetic field, does not simply increase.
Fig. 1. FDTD experimental configuration: Lightning base is located at
the center of the space and the distance between the base and a 10 m-
height overhead line is 50 m. Inclination angle θ and φ vary from ̶ 45˚
to +90˚ and 0˚ to +90˚, respectively.
(b) xy plane
(a) yz plane
Fig. 2. Return stroke current waveforms.
Fig. 4. Effect of the inclination angle θ on the peak induced
voltages: β = 1.00 and Tf = 1.0 μs.
Fig. 3. Induced voltages on the overhead line with the inclined
lightning of angle θ: β = 1.00 and Tf = 1.0 μs.
(b) ρe = 100 Ωm
(a) ρe = 0 Ωm
(c) ρe = 2000 Ωm
Fig. 6. Effect of the inclination angle φ on the peak induced
voltages: β = 1.00 and Tf = 1.0 μs.
Fig. 5. Induced voltages on the overhead line with the inclined
lightning of angle φ: β = 1.00 and Tf = 1.0 μs.
(b) ρe = 100 Ωm
(a) ρe = 0 Ωm
(c) ρe = 2000 Ωm
(a) Peak voltage versus angle θ
(b) Peak voltage versus angle φ
Fig. 8. Comparison of the effect of the lightning inclination on the
peak voltages between Tf =1.0 μs and 0.1 μs: β = 1.00.
Fig. 7. Comparisons of the induced voltage waveforms between
Tf =1.0μs and 0.1μs: β = 1.00 and θ = ̶ 45˚.
Fig. 8 (b) shows the effect of the angle φ on the induced
voltage for Tf = 1.0 and 0.1 μs. The peak voltages also become higher for Tf = 0.1 μs than 1.0 μs.
Considering the difference of the earth resistivity for the
waveforms in Fig. 7, the induced voltages at θ = ̶ 45˚ with Tf =
0.1 μs show almost the same amplitude of the peak. However,
the decay is different from each other, and similar to the result
of Tf = 1.0 μs. These mean that the waveform of the induced
voltage can be composed of the superposition of two
components: in the first one, at least with Tf = 0.1 μs, the
voltage is mainly induced by the lightning current without a
significant effect of the earth condition, and in the second one,
the voltage is result from an interaction between the current
and the earth.
D. Effect of return stroke velocity β
Fig. 9 illustrates the peak induced voltages for return
stroke velocity β = 0.33. Both results of Tf = 1.0 and 0.1 μs
shows almost the same manner to the result of β = 1.00 shown
in Fig. 8. The peak values show minor difference from the
result of β = 1.00.
IV. EARTH SURFACE CURRENT
The distribution and the effect of the earth current are
investigated by using the same models in the previous study.
Fig. 10 shows the earth surface current density just under the
overhead line (at the distances of 50 m from the lightning
channel base) when the lightning of β = 1.00 strikes the earth
for: (a) ρe = 0 Ωm, (b) ρe = 100 Ωm, and (c) ρe = 2000 Ωm.
Note that the direction of the current flow from the overhead
line toward the lightning base is considered as positive. It is
observed from the figure that the earth surface current
becomes larger when θ decreases. On the other hand, at t = 10
μs, the current of ρe = 0 Ωm at θ = ̶ 45˚ keeps higher value
while the current of ρe = 100 and 2000 Ωm become the same
value at θ = 0˚. This means that the resistivity of ρe = 100 and
2000 Ωm are more dominant for the distribution of the current
than inductive characteristic of the current path even if the
lightning is inclined.
Fig. 11 shows the peak values of the earth surface current
density. Note that solid black line and gray line indicate the
currents which are assumed to be distributed evenly in the
circular and hemispherical surface, respectively, i.e. the values
are:
)2( A/m2
1 ),1( A/m
2
1 2
2dist.
2
dist. r
Ir
I calhemisphericircular
The result for ρe = 0 Ωm at θ = 0˚ in Fig. 11 corresponds to
the black line. This means that, as expected, the earth current
flows only on the earth surface and scatters evenly for each
direction. On the other hand, the result for ρe = 2000 Ωm with
Tf = 1.0 μs in Fig. 11 (a) matches the gray line, indicating that
the current is distributed evenly on hemispherical surface due
to the earth resistivity as assumed above.
Fig. 9. Effect of the lightning inclination angle θ on the peak
voltages between Tf =1.0 μs and 0.1 μs: β = 0.33.
(a) Tf = 1.0 μs
(b) Tf = 0.1 μs
Fig. 11. The peak values of the earth surface current density caused
by the lightning inclined at θ = 0˚and ̶ 45˚: β = 1.00.
(a) ρe = 0 Ωm
(b) ρe = 100 Ωm
Fig. 10. The earth surface current density waveform in the earth
of ρe = 0, 100, and 2000 Ωm for the vertical lightning: β = 1.00.
(c) ρe = 2000 Ωm
Fig. 13. Peak differential voltage drop values between the earth
surface under the measuring point of the overhead line and the
surface of 50 m behind in y direction: β = 1.00.
Fig. 12. Horizontal differential voltage between the earth surface
under the measuring point of the overhead line and the surface of
50 m behind in y direction: β = 1.00, ρe = 2000 Ω.
The effect of the inclined lightning on the earth surface
current is also shown in Fig. 11. The current at θ = ̶ 45˚
becomes larger than that of θ = 0˚.
Fig. 12 shows the example of the horizontal differential
voltage between the earth surface under the measuring point of
the overhead line and the surface of 50 m behind in y direction.
Note that in ρe = 0 Ωm, there is no differential voltage between
the points. The ground potential significantly decreases due to
the voltage drop of the surface current, and the effect of angle
θ is clearly observed. The peak value of the differential
voltage is illustrated in Fig. 13. As expected, the voltage
decreases noticeably in the case of higher resistivity ρe = 2000
Ωm and shorter Tf = 0.1 μs. The earth voltage decrease occurs
when lightning strikes the lossy earth.
V. CONCLUSION
Lightning induced voltages on a distribution line have been
investigated by FDTD computations when the lightning
channel is angled and lossy earth is assumed. It is found that
when the lightning is inclined toward the line, it significantly
increases the induced voltage and the increase ratio is larger in
short rise time Tf. However, the inclination along the line
causes only a minor increment of the voltage. This
phenomenon is the same when the resistivity is relatively low.
For lightning with a slower return-stroke velocity, the effects
of angle, Tf and the earth resistivity on the voltage are almost
the same as those for β = 1.00. The inclined lightning also
affects the earth surface current and horizontal electric field
distribution, which transiently induces larger earth potential
drop in horizontal direction.
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